We shall now show how particle-hole symmetry can be used to establish localization along the lines of Jitomirskaya's arithmetic argument for the AMO [1]}. This argument heavily relies on the cosine-nature of the potential and requires non-resonant tunneling phases for \(A\)  and \(B\)  atoms of the potential. In particular, the argument does not seem to carry over easily to the case of chiral coupling.
We consider the Hamiltonian in the anti-chiral limit, for \(\theta =0\) , but take \(\phi =1/4\)  with plain cosine potential \(U(x)=\cos (2\pi x)\) . The Hamiltonian then reads in terms of \(K(\theta ):=1+e^{2\pi i \theta }(\tau +\tau ^*)\)
\(H_{w_0}(\theta ,\tfrac{1}{4},\vartheta ) = \begin{pmatrix} 0 &K(\theta )& w_0 U(\tfrac{x-\phi }{L}) & 0 \\K(\theta )^* &0 &0 & w_0U(\tfrac{x+\phi }{L}) \\w_0 U(\tfrac{x-\phi }{L}) &0&0 &K(\theta )\\0&w_0U(\tfrac{x+\phi }{L}) & K(\theta )^*&0 \end{pmatrix}.\) 
